Notes

The cellular automaton shown in the main text is purely two-dimensional. Experiments done on soap films since the 1980s indicate, however, that at least up to Reynolds numbers of several hundred, the patterns of flow around objects such as cylinders are almost identical to those seen in ordinary 3D fluids. The basic argument for Kolmogorov's k-5/3 result for the spectrum of turbulence is independent of dimension, but there are reasons to believe that in 2D eddies will tend to combine, so that after sufficiently long times only a small number of large eddies will be left. There is some evidence for this kind of process in the Earth's atmosphere, as well as in such phenomena as the Red Spot on Jupiter. At a microscopic level, there are some not completely unrelated issues in 2D about whether perturbations in a fluid made up of discrete molecules damp quickly enough to lead to ordinary viscosity. Formally, there is evidence that the Navier–Stokes equations in 2D might have a ∇2 Log[∇2] viscosity term, rather than a ∇2 one. But this effect, even if it is in fact present in principle, is almost certainly irrelevant on the scales of practical experiments.